Two Definitions of Limits, with Examples

Date: 05/11/98 at 05:23:25
From: Darryl Cain
Subject: Limit Theory
Dear Dr. Math,
I have seen many great articles from your homepage, and was wondering
if you could help me with a definition. I need to know what Limit
Theory is, and I can't find it anywhere.
I know it would be great to hear from you.
Thanks a lot,
Darryl Cain

Date: 05/11/98 at 15:09:27
From: Doctor Rob
Subject: Re: Limit Theory
I am not aware of a subject with the title "Limit Theory." One does
study limits when one takes Calculus. There are several kinds of
limits that are studied. One definition is like this:
Given a function f(x) from the real numbers to the real numbers,
and two real numbers a and L, we say that the limit of f(x) as x
approaches a is equal to L, and write
lim f(x) = L
x->a
if, for every epsilon > 0, no matter how small, there exists a
delta > 0 such that
|x - a| < delta implies that |f(x) - L| < epsilon.
You can see that this has to do with values of the function f(x) being
near L for all values of x near enough to a.
Example:
lim sin(x)/x = 1
x->0
because for every epsilon > 0, no matter how small, we can put
delta = sqrt(6*epsilon). Then, since x > sin(x) > x - x^3/3! for
x > 0, and since x - x^3/3! > sin(x) > x for x < 0, and sin(x) = x for
x = 0, we have that:
|sin(x) - x| <= |x^3/3!| = (|x|^3)/6
Then whenever |x| = |x - 0| < delta,
|sin(x)/x - 1| <= (|x|^2)/6 < (delta^2)/6 <= epsilon
Another definition is like this:
Given an infinite sequence of real numbers:
{x(n): n = 1, 2, 3, ...},
and a real number L, we say that the limit of x(n) as n approaches
infinity is equal to L, and write:
lim x(n) = L
n->infinity
if, for every epsilon > 0, no matter how small, there is a natural
number N > 0 such that n > N implies that |x(n) - L| < epsilon.
You can see that this has to do with the value of terms of the
sequence being near L for all large enough values of n.
Example:
lim (n + 1)/(2*n + 1) = 1/2
n->infinity
because, for every epsilon > 0, no matter how small, we can put
N > 1/(4*epsilon). Then whenever n > N > 1/(4*epsilon),
epsilon > 1/(4*n) > 1/(4*n + 2)
= |(n + 1)/(2*n + 1) - 1/2| = |x(n) - L|
-Doctor Rob, The Math Forum
Check out our web site! http://mathforum.org/dr.math/